 Solution

The vector triple product identity gives $\mathbf{n}\times (\mathbf{n}\times \mathbf{x})=\mathbf{n}(\mathbf{n}\cdot \mathbf{x})-\mathbf{x}$, and henceThus $\Phi$ is the identity on the orthogonal complement ${\mathbf{n}}^{\perp }$ and multiplies the component parallel to $\mathbf{n}$ by $\alpha$. It is invertible exactly when $\alpha \ne 0$, and then